Asymptotically periodic behavior of solutions of fractional evolution equations of order 1 < α < 2

Abstract

In this paper we investigate the asymptotically periodic behavior of solutions of fractional evolution equations of order 1 &lt; α &lt; 2 and in particular existence and uniqueness results are established. Two examples are given to illustrate our results.

Asymptotically periodic behavior of solutions to fractional non-instantaneous impulsive semilinear differential inclusions with sectorial operators

“…In the first result, we apply a fixed point theorem for contraction multivalued functions, and, in the second result, we use a compactness criterion in the space of bounded piecewise continuous functions defined on the unbounded interval J = [0, ∞). Our work generalizes much recent work such as [18][19][20] to the case where there are impulse effects, and the right-hand side is a multivalued function.…”

“…with D(A) = H 2 ( ) ∩ H 1 0 ( ). It is known that (see [19]) A is a sectorial operator of type {M, ϕ, α, μ} with μ = -1 and L = 3. Let Z be a non-empty, compact and convex subset of E, υ = sup z∈Z z .…”

“…Therefore, the concept of an asymptotically periodic solution for fractional differential equations or inclusions is introduced and discussed in much work. For example, in [12,[15][16][17], the authors considered semilinear differential equations of order α ∈ (0, 1) generated by a C 0semigroup, while the papers [18][19][20] addressed semilinear differential equations of order α ∈ (0, 1) generated by sectorial operators. Moreover, the asymptotically periodic solutions for delayed fractional differential equations with almost sectorial operator of order α ∈ (0, 1) are examined in [21].…”

“…To clarify the advantage of this study, we mention that two methods have been provided to demonstrate the existence of S-asymptotic ω-periodic solutions for semilinear fractional differential inclusions in the presence of non-instantaneous impulse effects, and in which the nonlinear part is a multivalued function, and the linear part is a sectorial operator. Moreover, the technique presented in this paper can be used to generalize the work in [12,[15][16][17][18][19][20][21][23][24][25] to the case where the linear part is a sectorial operator, the nonlinear part is a multivalued function, and there is impulse effects. In addition, Problem (1) can be investigated on time scales using the arguments in [1], and using the arguments in [3,6,8], one can examine the asymptotic periodic solutions for Problem (1) when the Caputo derivative is replaced by the ψ-Caputo derivative, ψ-RL derivative, Atangana-Baleanu derivative or p-Laplacian operator.…”

Extremal Solutions for a Class of Tempered Fractional Turbulent Flow Equations in a Porous Medium

“…Particularly, if the model is of fractional order, then it can describe turbulent flow in a porous medium [5][6][7][8][9][10]. On the contrary, fractional-order derivative has nonlocal characteristics; based on this property, the fractional differential equation can also interpret many abnormal phenomena that occur in applied science and engineering, such as viscoelastic dynamical phenomena [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29], advection-dispersion process in anomalous diffusion [30][31][32][33][34], and bioprocesses with genetic attribute [35,36]. As a powerful tool of modeling the above phenomena, in recent years, the fractional calculus theory has been perfected gradually by many researchers, and various different types of fractional derivatives were studied, such as Riemann-Liouville derivatives [16,, Hadamardtype derivatives [63][64][65][66][67][68][69][70][71], Katugampola-Caputo derivatives [72], conformable derivatives…”